A third estimate, in this case based on the electric picture, for \(\left<\mathbf {E}\cdot \mathbf {B}\right>\)  has recently been presented in Ref. [1]}.
This estimate is based on the gradient expansion formalism (GEF) developed in Ref. [2]}, which describes the evolution of all relevant background quantities during axion inflation, \(\left<\mathbf {E}^2\right>\) , \(\left<\mathbf {B}^2\right>\) , \(\left<\mathbf {E}\cdot \mathbf {B}\right>\) , etc., in terms of bilinear scalar functions that are constructed from the vector fields \(\mathbf {E}\)  and \(\mathbf {B}\)  in position space (see also Ref. [3]} for related earlier work).
If one explicitly specifies the scalar potential \(V\)  and initial conditions, this approach allows one to explicitly track the dynamics of axion inflation, even in the presence of nonlinear backreaction and nonperturbative fermion production, with unprecedented accuracy.
However, for the purposes of the present paper, we are rather interested in more general, model-independent predictions.
We will therefore not use the results of Ref. [2]} and implement the gradient expansion formalism for a specific model.
Instead, we will resort to the model-independent results derived in Ref. [1]}, which are based on the assumption that \(\xi \)  and \(H\)  only vary slowly during inflation.
Towards the end of inflation, which is the point in time we are most interested in, this assumption becomes violated.
Still, the analysis in Ref. [1]} was able to show that the model-independent results continue to represent a good approximation, typically within less than one order of magnitude of the exact results and in better agreement with the exact results than the equilibrium and maximal estimates.
